Which-path problem for one and two particles with two degrees of freedom and a relation between transverse spatial structure and group velocity of light

Quantum mechanics, optics and indeed any wave theory exhibits the phenomenon of interference. In this thesis we present two problems investigating interference due to indistinguishable alternatives and a mostly unrelated investigation into the free space propagation speed of light pulses in particular spatial modes. In chapter 1 we introduce the basic properties of the electromagnetic field needed for the subsequent chapters. In chapter 2 we review the properties of interference using the beam splitter and the Mach-Zehnder interferometer. In particular we review what happens when one of the paths of the interferometer is marked in some way so that the particle having traversed it contains information as to which path it went down (to be followed up in chapter 3) and we review Hong-Ou-Mandel interference at a beam splitter (to be followed up in chapter 5). In chapter 3 we present the first of the interference problems. This consists of a nested Mach-Zehnder interferometer in which each of the free space propagation segments are weakly marked by mirrors vibrating at different frequencies [1]. The original experiment drew the conclusions that the photons followed disconnected paths. We partition the description of the light in the interferometer according to the number of paths it contains which-way information about and reinterpret the results reported in [1] in terms of the interference of paths spatially connected from source to detector. In chapter 4 we briefly review optical angular momentum, entanglement and spontaneous parametric down conversion. These concepts feed into chapter 5 in which we present the second of the interference problems namely Hong-Ou-Mandel interference with particles possessing two degrees of freedom. We analyse the problem in terms of exchange symmetry for both boson and fermion pairs and show that the particle statistics at a beam splitter can be controlled for suitably chosen states. We propose an experimental test of these ideas using orbital angular momentum entangled photons. In chapter 6 we look at the effect that the transverse spatial structure of the mode that a pulse of light is excited in has on its group velocity. We show that the resulting group velocity is slower than the speed of light in vacuum for plane waves and that this reduction in the group velocity is related to the spread in the wave vectors required to create the transverse spatial structure. We present experimental results of the measurement of this slowing down using Hong-Ou-Mandel interference.

Pollinator community responses to the spatial population structure of wild plants: A pan-European approach

Land-use changes can alter the spatial population structure of plant species, which may in turn affect the attractiveness of flower aggregations to different groups of pollinators at different spatial scales. To assess how pollinators respond to spatial heterogeneity of plant distributions and whether honeybees affect visitation by other pollinators we used an extensive data set comprising ten plant species and their flower visitors from five European countries. In particular we tested the hypothesis that the composition of the flower visitor community in terms of visitation frequencies by different pollinator groups were affected by the spatial plant population structure, viz. area and density measures, at a within-population (‘patch’) and among-population (‘population’) scale. We found that patch area and population density were the spatial variables that best explained the variation in visitation frequencies within the pollinator community. Honeybees had higher visitation frequencies in larger patches, while bumblebees and hoverflies had higher visitation frequencies in sparser populations. Solitary bees had higher visitation frequencies in sparser populations and smaller patches. We also tested the hypothesis that honeybees affect the composition of the pollinator community by altering the visitation frequencies of other groups of pollinators. There was a positive relationship between visitation frequencies of honeybees and bumblebees, while the relationship with hoverflies and solitary bees varied (positive, negative and no relationship) depending on the plant species under study. The overall conclusion is that the spatial structure of plant populations affects different groups of pollinators in contrasting ways at both the local (‘patch’) and the larger (‘population’) scales and, that honeybees affect the flower visitation by other pollinator groups in various ways, depending on the plant species under study. These contrasting responses emphasize the need to investigate the entire pollinator community when the effects of landscape change on plant–pollinator interactions are studied.

Competition may explain the fine-scale spatial patterns and genetic structure of two co-occurring plant congeners

1. The spatial distribution of individual plants within a population and the population’s genetic structure are determined by several factors, like dispersal, reproduction mode or biotic interactions. The role of interspecific interactions in shaping the spatial genetic structure of plant populations remains largely unknown. 2. Species with a common evolutionary history are known to interact more closely with each other than unrelated species due to the greater number of traits they share. We hypothesize that plant interactions may shape the fine genetic structure of closely related congeners. 3. We used spatial statistics (georeferenced design) and molecular techniques (ISSR markers) to understand how two closely related congeners, Thymus vulgaris (widespread species) and T. loscosii (narrow endemic) interact at the local scale. Specific cover, number of individuals of both study species and several community attributes were measured in a 10 × 10 m plot. 4. Both species showed similar levels of genetic variation, but differed in their spatial genetic structure. Thymus vulgaris showed spatial aggregation but no spatial genetic structure, while T. loscosii showed spatial genetic structure (positive genetic autocorrelation) at short distances. The spatial pattern of T. vulgaris’ cover showed significant dissociation with that of T. loscosii. The same was true between the spatial patterns of the cover of T. vulgaris and the abundance of T. loscosii and between the abundance of each species. Most importantly, we found a correlation between the genetic structure of T. loscosii and the abundance of T. vulgaris: T. loscosii plants were genetically more similar when they were surrounded by a similar number of T. vulgaris plants. 5. Synthesis. Our results reveal spatially complex genetic structures of both congeners at small spatial scales. The negative association among the spatial patterns of the two species and the genetic structure found for T. loscosii in relation to the abundance of T. vulgaris indicate that competition between the two species may account for the presence of adapted ecotypes of T. loscosii to the abundance of a competing congeneric species. This suggests that the presence and abundance of close congeners can influence the genetic spatial structure of plant species at fine scales.

Measurement of genetic structure within populations using Moran's spatial autocorrelation statistics.

Spatial structure of genetic variation within populations, an important interacting influence on evolutionary and ecological processes, can be analyzed in detail by using spatial autocorrelation statistics. This paper characterizes the statistical properties of spatial autocorrelation statistics in this context and develops estimators of gene dispersal based on data on standing patterns of genetic variation. Large numbers of Monte Carlo simulations and a wide variety of sampling strategies are utilized. The results show that spatial autocorrelation statistics are highly predictable and informative. Thus, strong hypothesis tests for neutral theory can be formulated. Most strikingly, robust estimators of gene dispersal can be obtained with practical sample sizes. Details about optimal sampling strategies are also described.

The spatial effect of intra-metropolitan agglomeration economies

This study deals with the role of spatial accessibility to agglomeration economies in the change in spatial structure of industrial employment for the case of the Barcelona Metropolitan Region of Barcelona (BMR). Using the growth in gross density of municipal employment between 1986 and 1996 for seven manufacturing industries as an indicator of changes in the spatial structure of employment, an exploration is made of the spatial impact of agglomeration economies operating on a local scale the municipality and three areas 5, 8 and 12 kilometres away surrounding the municipality itself - , agglomeration economies emerging from CBD and the main specialised subcentres in the region, and the network economies associated with the total jobs in the region, access to which depends on the distance from the main transport infrastructures

Spatial patterns of preconsolidation pressure and soil moisture along transects in two directions under coffee

Information on the spatial structure of soil physical and structural properties is needed to evaluate the soil quality. The purpose of this study was to investigate the spatial behavior of preconsolidation pressure and soil moisture in six transects, three selected along and three across coffee rows, at three different sites under different tillage management systems. The study was carried out on a farm, in Patrocinio, state of Minas Gerais, in the Southeast of Brazil (18 º 59 ' 15 '' S; 46 º 56 ' 47 '' W; 934 m asl). The soil type is a typic dystrophic Red Latosol (Acrustox) and consists of 780 g kg-1 clay; 110 g kg-1 silt and 110 g kg-1 sand, with an average slope of 3 %. Undisturbed soil cores were sampled at a depth of 0.10-0.13 m, at three different points within the coffee plantation: (a) from under the wheel track, where equipment used in farm operations passes; (b) in - between tracks and (c) under the coffee canopy. Six linear transects were established in the experimental area: three transects along and three across the coffee rows. This way, 161 samples were collected in the transect across the coffee rows, from the three locations, while 117 samples were collected in the direction along the row. The shortest sampling distance in the transect across the row was 4 m, and 0.5 m for the transect along the row. No clear patterns of the preconsolidation pressure values were observed in the 200 m transect. The results of the semivariograms for both variables indicated a high nugget value and short range for the studied parameters of all transects. A cyclic pattern of the parameters was observed for the across-rows transect. An inverse relationship between preconsolidation pressure and soil moisture was clearly observed in the samples from under the track, in both directions.

Effect of spatial processes on the evolution of ecological niches and dispersal in metacommunity systems

RésuméLa coexistence de nombreuses espèces différentes a de tout temps intrigué les biologistes. La diversité et la composition des communautés sont influencées par les perturbations et l'hétérogénéité des conditions environnementales. Bien que dans la nature la distribution spatiale des conditions environnementales soit généralement autocorrélée, cet aspect est rarement pris en compte dans les modèles étudiant la coexistence des espèces. Dans ce travail, nous avons donc abordé, à l'aide de simulations numériques, la coexistence des espèces ainsi que leurs caractéristiques au sein d'un environnement autocorrélé.Afin de prendre en compte cet élément spatial, nous avons développé un modèle de métacommunauté (un ensemble de communautés reliées par la dispersion des espèces) spatialement explicite. Dans ce modèle, les espèces sont en compétition les unes avec les autres pour s'établir dans un nombre de places limité, dans un environnement hétérogène. Les espèces sont caractérisées par six traits: optimum de niche, largeur de niche, capacité de dispersion, compétitivité, investissement dans la reproduction et taux de survie. Nous nous sommes particulièrement intéressés à l'influence de l'autocorrélation spatiale et des perturbations sur la diversité des espèces et sur les traits favorisés dans la métacommunauté. Nous avons montré que l'autocorrélation spatiale peut avoir des effets antagonistes sur la diversité, en fonction du taux de perturbations considéré. L'influence de l'autocorrélation spatiale sur la capacité de dispersion moyenne dans la métacommunauté dépend également des taux de perturbations et survie. Nos résultats ont aussi révélé que de nombreuses espèces avec différents degrés de spécialisation (i.e. différentes largeurs de niche) peuvent coexister. Toutefois, les espèces spécialistes sont favorisées en absence de perturbations et quand la dispersion est illimitée. A l'opposé, un taux élevé de perturbations sélectionne des espèces plus généralistes, associées avec une faible compétitivité.L'autocorrélation spatiale de l'environnement, en interaction avec l'intensité des perturbations, influence donc de manière considérable la coexistence ainsi que les caractéristiques des espèces. Ces caractéristiques sont à leur tour souvent impliquées dans d'importants processus, comme le fonctionnement des écosystèmes, la capacité des espèces à réagir aux invasions, à la fragmentation de l'habitat ou aux changements climatiques. Ce travail a permis une meilleure compréhension des mécanismes responsables de la coexistence et des caractéristiques des espèces, ce qui est crucial afin de prédire le devenir des communautés naturelles dans un environnement changeant.AbstractUnderstanding how so many different species can coexist in nature is a fundamental and long-standing question in ecology. Community diversity and composition are known to be influenced by heterogeneity in environmental conditions and disturbance. Though in nature the spatial distribution of environmental conditions is frequently autocorrelated, this aspect is seldom considered in models investigating species coexistence. In this work, we thus addressed several questions pertaining to species coexistence and composition in spatially autocorrelated environments, with a numerical simulations approach.To take into account this spatial aspect, we developed a spatially explicit model of metacommunity (a set of communities linked by dispersal of species). In this model, species are trophically equivalent, and compete for space in a heterogeneous environment. Species are characterized by six life-history traits: niche optimum, niche breadth, dispersal, competitiveness, reproductive investment and survival rate. We were particularly interested in the influence of environmental spatial autocorrelation and disturbance on species diversity and on the traits of the species favoured in the metacommunity. We showed that spatial autocorrelation can have antagonistic effects on diversity depending on disturbance rate. Similarly, spatial autocorrelation interacted with disturbance rate and survival rate to shape the mean dispersal ability observed in the metacommunity. Our results also revealed that many species with various degrees of specialization (i.e. different niche breadths) can coexist together. However specialist species were favoured in the absence of disturbance, and when dispersal was unlimited. In contrast, high disturbance rate selected for more generalist species, associated with low competitive ability.The spatial structure of the environment, together with disturbance and species traits, thus strongly impacts species diversity and, more importantly, species composition. Species composition is known to affect several important metacommunity properties such as ecosystem functioning, resistance and reaction to invasion, to habitat fragmentation and to climate changes. This work allowed a better understanding of the mechanisms responsible for species composition, which is of crucial importance to predict the fate of natural metacommunities in changing environments

Time and space: genetic structure of the common sea urchin paracentrotus lividus in the south-eastern iberian coast

Spatio-temporal variability in settlement and recruitment, high mortality during the first life-history stages, and selection may determine the genetic structure of cohorts of long-lived marine invertebrates at small scales. We conducted a spatial and temporal analysis of the common Mediterranean Sea urchin Paracentrotus lividus to determine the genetic structure of cohorts at different scales. In Tossa de Mar (NW Mediterranean), recruitment was followed over 5 consecutive springs (2006-2010). In spring 2008, recruits and two-year-old individuals were collected at 6 locations along East and South Iberian coasts separated from 200 to over 1,100 km. All cohorts presented a high genetic diversity based on a fragment of mtCOI. Our results showed a marked genetic homogeneity in the temporal monitoring and a low degree of spatial structure in 2006. In 2008, coupled with an abnormality in the usual circulation patterns in the area, the genetic structure of the southern populations studied changed markedly, with arrival of many private haplotypes. This fact highlights the importance of point events in renewing the genetic makeup of populations, which can only be detected through analysis of the cohort structure coupling temporal and spatial perspectives.

GPS receivers for georeferencing of spatial variability of soil attributes

The characterization of the spatial variability of soil attributes is essential to support agricultural practices in a sustainable manner. The use of geostatistics to characterize spatial variability of these attributes, such as soil resistance to penetration (RP) and gravimetric soil moisture (GM) is now usual practice in precision agriculture. The result of geostatistical analysis is dependent on the sample density and other factors according to the georeferencing methodology used. Thus, this study aimed to compare two methods of georeferencing to characterize the spatial variability of RP and GM as well as the spatial correlation of these variables. Sampling grid of 60 points spaced 20 m was used. For RP measurements, an electronic penetrometer was used and to determine the GM, a Dutch auger (0.0-0.1 m depth) was used. The samples were georeferenced using a GPS navigation receiver, Simple Point Positioning (SPP) with navigation GPS receiver, and Semi-Kinematic Relative Positioning (SKRP) with an L1 geodetic GPS receiver. The results indicated that the georeferencing conducted by PPS did not affect the characterization of spatial variability of RP or GM, neither the spatial structure relationship of these attributes.

Spatial covariation of Azotobacter abundance and soil properties: A case study using the wavelet transform

The soil microflora is very heterogeneous in its spatial distribution. The origins of this heterogeneity and its significance for soil function are not well understood. A problem for understanding spatial variation better is the assumption of statistical stationarity that is made in most of the statistical methods used to assess it. These assumptions are made explicit in geostatistical methods that have been increasingly used by soil biologists in recent years. Geostatistical methods are powerful, particularly for local prediction, but they require the assumption that the variability of a property of interest is spatially uniform, which is not always plausible given what is known about the complexity of the soil microflora and the soil environment. We have used the wavelet transform, a relatively new innovation in mathematical analysis, to investigate the spatial variation of abundance of Azotobacter in the soil of a typical agricultural landscape. The wavelet transform entails no assumptions of stationarity and is well suited to the analysis of variables that show intermittent or transient features at different spatial scales. In this study, we computed cross-variograms of Azotobacter abundance with the pH, water content and loss on ignition of the soil. These revealed scale-dependent covariation in all cases. The wavelet transform also showed that the correlation of Azotobacter abundance with all three soil properties depended on spatial scale, the correlation generally increased with spatial scale and was only significantly different from zero at some scales. However, the wavelet analysis also allowed us to show how the correlation changed across the landscape. For example, at one scale Azotobacter abundance was strongly correlated with pH in part of the transect, and not with soil water content, but this was reversed elsewhere on the transect. The results show how scale-dependent variation of potentially limiting environmental factors can induce a complex spatial pattern of abundance in a soil organism. The geostatistical methods that we used here make assumptions that are not consistent with the spatial changes in the covariation of these properties that our wavelet analysis has shown. This suggests that the wavelet transform is a powerful tool for future investigation of the spatial structure and function of soil biota. (c) 2006 Elsevier Ltd. All rights reserved.

The Representative Soil Sampling Scheme of England and Wales: the spatial variation of topsoil nutrient status and pH between 1971 and 2001

The Representative Soil Sampling Scheme (RSSS) has monitored the soil of agricultural land in England and Wales since 1969. Here we describe the first spatial analysis of the data from these surveys using geostatistics. Four years of data (1971, 1981, 1991 and 2001) were chosen to examine the nutrient (available K, Mg and P) and pH status of the soil. At each farm, four fields were sampled; however, for the earlier years, coordinates were available for the farm only and not for each field. The averaged data for each farm were used for spatial analysis and the variograms showed spatial structure even with the smaller sample size. These variograms provide a reasonable summary of the larger scale of variation identified from the data of the more intensively sampled National Soil Inventory. Maps of kriged predictions of K generally show larger values in the central and southeastern areas (above 200 mg L-1) and an increase in values in the west over time, whereas Mg is fairly stable over time. The kriged predictions of P show a decline over time, particularly in the east, and those of pH show an increase in the east over time. Disjunctive kriging was used to examine temporal changes in available P using probabilities less than given thresholds of this element. The RSSS was not designed for spatial analysis, but the results show that the data from these surveys are suitable for this purpose. The results of the spatial analysis, together with those of the statistical analyses, provide a comprehensive view of the RSSS database as a basis for monitoring the soil. These data should be taken into account when future national soil monitoring schemes are designed.

A geostatistical analysis of the spatial variation of soil mineral nitrogen and potentially available nitrogen within an arable field.

The technology for site-specific applications of nitrogen (N) fertilizer has exposed a gap in our knowledge about the spatial variation of soil mineral N, and that which will become available during the growing season within arable fields. Spring mineral N and potentially available N were measured in an arable field together with gravimetric water content, loss on ignition, crop yield, percentages of sand, silt, and clay, and elevation to describe their spatial variation geostatistically. The areas with a larger clay content had larger values of mineral N, potentially available N, loss on ignition and gravimetric water content, and the converse was true for the areas with more sandy soil. The results suggest that the spatial relations between mineral N and loss on ignition, gravimetric water content, soil texture, elevation and crop yield, and between potentially available N and loss on ignition and silt content could be used to indicate their spatial patterns. Variable-rate nitrogen fertilizer application would be feasible in this field because of the spatial structure and the magnitude of variation of mineral N and potentially available N.

Toward a combined seasonal weather and crop productivity forecasting system: Determination of the working spatial scale

A methodology is presented for the development of a combined seasonal weather and crop productivity forecasting system. The first stage of the methodology is the determination of the spatial scale(s) on which the system could operate; this determination has been made for the case of groundnut production in India. Rainfall is a dominant climatic determinant of groundnut yield in India. The relationship between yield and rainfall has been explored using data from 1966 to 1995. On the all-India scale, seasonal rainfall explains 52% of the variance in yield. On the subdivisional scale, correlations vary between variance r(2) = 0.62 (significance level p < 10(-4)) and a negative correlation with r(2) = 0.1 (p = 0.13). The spatial structure of the relationship between rainfall and groundnut yield has been explored using empirical orthogonal function (EOF) analysis. A coherent, large-scale pattern emerges for both rainfall and yield. On the subdivisional scale (similar to 300 km), the first principal component (PC) of rainfall is correlated well with the first PC of yield (r(2) = 0.53, p < 10(-4)), demonstrating that the large-scale patterns picked out by the EOFs are related. The physical significance of this result is demonstrated. Use of larger averaging areas for the EOF analysis resulted in lower and (over time) less robust correlations. Because of this loss of detail when using larger spatial scales, the subdivisional scale is suggested as an upper limit on the spatial scale for the proposed forecasting system. Further, district-level EOFs of the yield data demonstrate the validity of upscaling these data to the subdivisional scale. Similar patterns have been produced using data on both of these scales, and the first PCs are very highly correlated (r(2) = 0.96). Hence, a working spatial scale has been identified, typical of that used in seasonal weather forecasting, that can form the basis of crop modeling work for the case of groundnut production in India. Last, the change in correlation between yield and seasonal rainfall during the study period has been examined using seasonal totals and monthly EOFs. A further link between yield and subseasonal variability is demonstrated via analysis of dynamical data.

Power properties if invariant tests for spatial autocorrelation in linear regression

This paper derives some exact power properties of tests for spatial autocorrelation in the context of a linear regression model. In particular, we characterize the circumstances in which the power vanishes as the autocorrelation increases, thus extending the work of Krämer (2005). More generally, the analysis in the paper sheds new light on how the power of tests for spatial autocorrelation is affected by the matrix of regressors and by the spatial structure. We mainly focus on the problem of residual spatial autocorrelation, in which case it is appropriate to restrict attention to the class of invariant tests, but we also consider the case when the autocorrelation is due to the presence of a spatially lagged dependent variable among the regressors. A numerical study aimed at assessing the practical relevance of the theoretical results is included